Nontotient
In number theory, a nontotient is a positive integer n'' which is not a totient number: it is not in the range of Euler's totient function φ, that is, the equation φ(''x) = n'' has no solution ''x. In other words, n'' is a nontotient if there is no integer ''x that has exactly n'' coprimes below it. All odd numbers are nontotients, except 1, since it has the solutions ''x = 1 and x'' = 2. The first few even nontotients are 12, 22, 2X, 32, 42, 52, 58, 62, 64, 72, 76, 7X, 82, 96, 9X, X2, X4, E2, EX, 102, 108, 10X, 112, 122, 126, 132, 136, 138, 142, 14X, 152, 15X, 162, 172, 176, 178, 182, 184, 186, 188, 192, 196, 1X2, 1XX, 1E2, 1E8, 1EX, 202, 20X, 212, 214, 218, 222, 226, 22X, 232, 23X, 242, 244, 252, 256, 262, 264, 26X, 272, 274, 282, 286, 28X, 292, 296, 298, 29X, 2X2, 2X4, 2X6, 2E2, 2E6, 2E8, 302, 304, 312, 316, 31X, 322, 332, 334, 336, 342, 344, 348, 352, 354, 366, 36X, 370, 372, 37X, 382, 384, 386, 38X, 392, 398, 39X, 3X2, 3X6, 3E2, 3E8, 3EX, 402, 406, 412, 416, 418, 422, 424, 428, 42X, 432, 43X, 442, 444, 44X, 452, 458, 462, 466, 472, 476, 478, 47X, 482, 486, 488, 492, 49X, 4X2, 4X6, 4XX, 4E2, 4E6, 502, 504, 508, 50X, 512, 518, 522, 524, 528, 52X, 532, 536, 53X, 542, 546, 54X, 552, 558, 55X, 562, 566, 56X, 570, 572, 57X, 582, 588, 592, 596, 5X2, 5X4, 5X6, 5XX, 5E2, 602, 604, 606, 608, 60X, 612, 622, 624, 626, 62X, 632, 638, 642, 644, 64X, 652, 65X, 662, 666, 668, 670, 672, 676, 67X, 682, 684, 688, 692, 696, 6X2, 6X4, 6X8, 6XX, 6E2, 6E6, 6EX, 702, 706, 712, 716, 718, 722, 724, 72X, 732, 738, 73X, 742, 748, 74X, 752, 756, 758, 75X, 762, 764, 772, 776, 778, 782, 786, 78X, 792, 79X, 7X2, 7X4, 7X6, 7XX, 7E2, 7E6, 7E8, 802, 804, 806, 810, 812, 81X, 822, 826, 832, 836, 83X, 842, 844, 846, 848, 852, 856, 858, 862, 86X, 872, 876, 878, 87X, 882, 884, 886, 888, 892, 896, 898, 89X, 8X2, 8E2, 8EX, 902, 910, 912, 916, 922, 924, 92X, 932, 934, 936, 938, 93X, 942, 944, 946, 94X, 952, 956, 962, 966, 96X, 972, 976, 978, 97X, 982, 984, 98X, 992, 996, 998, 99X, 9X2, 9X4, 9E2, 9E6, X02, X04, X08, X12, X18, X1X, X22, X24, X2X, X32, X42, X46, X48, X52, X54, X56, X58, X62, X64, X66, X70, X72, X78, X7X, X82, X8X, X92, X96, X98, XX2, XX4, XE2, XE4, XE8, E02, E04, E06, E0X, E12, E16, E18, E22, E26, E32, E38, E3X, E42, E46, E4X, E52, E54, E56, E5X, E62, E72, E76, E7X, E82, E84, E86, E88, E8X, E92, E98, E9X, EX2, EX6, EXX, EE2, EEX, ... An even nontotient may be one more than a prime number, but never one less, since all numbers below a prime number are, by definition, coprime to it. To put it algebraically, for p prime: φ(''p) = p'' − 1. Also, a pronic number ''n(n'' − 1) is certainly not a nontotient if ''n is prime since φ(p''2) = ''p(p'' − 1). If a natural number ''n is a totient, it can be shown that n''×2''k is a totient for all natural number k''. There are infinitely many even nontotient numbers: indeed, there are infinitely many distinct primes ''p such that all numbers of the form 2''a'p'' are nontotient, and every odd number has an even multiple which is a nontotient. Most numbers end with 2 are nontotients, except 2 itself, the first counterexample is 92 = φ(X1) = φ(182), in fact, all numbers end with 2 not of the form (p−1)''p'k'' with p'' prime and ''k ≥ 1 are nontotients, (see OEIS sequence A063668 for the counterexamples) (besides, numbers end with 2 except 2 itself cannot be totients of squarefree numbers), besides, all even nontotients < 58 except 2X end with 2. Category:Pages